qu.1.topic=a^log[z](x)@

qu.1.1.question=<p>Simplify $Q.</p>@
qu.1.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.1.1.allow2d=1@
qu.1.1.maple_answer=$ANS@
qu.1.1.type=formula@
qu.1.1.mode=Maple@
qu.1.1.name=e^log(f^g)@
qu.1.1.comment=<p>$Q</p>
<p>= $Step1</p>
<p>= $Step2</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=range(2,15);
$n=range(2,10);
$f=switch(rint(4),"x^($n)","sin(x)","cos(x)","tan(x)");
$g="x^$n";
condition: ne($f,$g);
$ANS="($f)^($g)";
$Q=maple("printf(MathML[ExportPresentation](($a)^(($g)*'log[$a]'($f))))");
$Step1=maple("printf(MathML[ExportPresentation](($a)^('log[$a]'($ANS))))");
$Step2=mathml("$ANS");@
qu.1.1.uid=d4314117-6982-4b34-b12d-76f236645dec@
qu.1.1.info=  Author=Jack Weiner, Gord Clemet;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Logs and Exponents;
  Sub-Topic=Logarithm rules;
@

qu.1.2.question=<p>Simplify $Q.</p>@
qu.1.2.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.1.2.allow2d=1@
qu.1.2.maple_answer=$ANS@
qu.1.2.type=formula@
qu.1.2.mode=Maple@
qu.1.2.name=e^logx@
qu.1.2.comment=<p>Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>a</mi><mrow><msub><mi mathvariant='normal'>log</mi><mrow><mi>a</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=range(2,15);
$n=range(1,10);
$f=switch(rint(4),"x^($n)","sin(x)","cos(x)","tan(x)");
$ANS="$f";
$Q=maple("printf(MathML[ExportPresentation]($a^('log[$a]'($f))))");@
qu.1.2.uid=a60a79ad-85ca-4580-a6c9-2bf8efeabca9@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Logarithm rules;
@

qu.2.topic=log(a^x)@

qu.2.1.question=<p>Simplify $Q.</p>@
qu.2.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=$ANS@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=log[a](a^x)@
qu.2.1.comment=<p>Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi mathvariant='normal'>log</mi><mrow><mi>a</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><msup><mi>a</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=range(2,15);
$n=range(1,10);
$f=switch(rint(4),"x^($n)","sin(x)","cos(x)","tan(x)");
$ANS="$f";
$Q=maple("printf(MathML[ExportPresentation]('log[$a]'($a^($f))))");@
qu.2.1.uid=babc6319-e373-4178-992c-ce4e1e0f1dec@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Logarithm rules;
@

qu.3.topic=derivatives@

qu.3.1.question=<p>Given $F, find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>. (Enter exp(x) for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow></math>.)</p>@
qu.3.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=$ANS@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=derivatives@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=range(-5,5);
condition:ne($a,0);
$b=switch(rint(5),2,3,4,5,10);
$z=rint(4);
$f=switch($z,"ln(($a)*x)","exp(($a)*x)","ln(abs(($a)*x))","($b)^x");
$ANS=switch($z, "diff($f,x)", "diff($f,x)", "1/x" ,"diff($f,x)");
$F=maple("printf(MathML[ExportPresentation](y=$f))");@
qu.3.1.uid=3f0d9280-249f-46a6-850b-71f930329bca@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Logs and Exponents;
  Sub-Topic=Derivatives;
@

qu.4.topic=domain,range@

qu.4.1.mode=Inline@
qu.4.1.name=Log(x)DomRange@
qu.4.1.comment=<p>$plot</p>
<p>Does your answer for domain and range match the graph?</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(0,5);
$b=rint(2,10);
$z=rint(2);
$f=switch($z,"log[($b)](x-($a))","log[($b)](($a)-x)");
$fg=switch($z,"log[($b)](x-($a))","log[($b)](($a)-x)");
$R="(-infinity,infinity)";
$D=switch($z,"($a,infinity)","(-infinity,$a)");
$F=maple("printf(MathML[ExportPresentation](y='$f'))");
$plot=switch($z,plotmaple("plot($fg,x=-1..($a)+($b)+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'"),plotmaple("plot($fg,x=-($a)-($b)-1..($a)+1,thickness=2),
plotdevice='gif', plotoptions='height=250,width=250'"));@
qu.4.1.uid=525055df-37b1-400c-aa9b-09b585d1b525@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Domain and Range;
@
qu.4.1.weighting=1,1,1@
qu.4.1.numbering=alpha@
qu.4.1.part.1.name=sro_id_1@
qu.4.1.part.1.maple_answer=show("$D");@
qu.4.1.part.1.editing=useHTML@
qu.4.1.part.1.question=(Unset)@
qu.4.1.part.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.1.part.1.mode=Maple@
qu.4.1.part.1.allow2d=0@
qu.4.1.part.1.plot=@
qu.4.1.part.1.maple=grade("$RESPONSE","$D");
@
qu.4.1.part.1.type=maple@
qu.4.1.part.2.name=sro_id_2@
qu.4.1.part.2.maple_answer=show("$R");@
qu.4.1.part.2.editing=useHTML@
qu.4.1.part.2.question=(Unset)@
qu.4.1.part.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.1.part.2.mode=Maple@
qu.4.1.part.2.allow2d=0@
qu.4.1.part.2.plot=@
qu.4.1.part.2.maple=grade("$RESPONSE","$R");@
qu.4.1.part.2.type=maple@
qu.4.1.part.3.grader=exact@
qu.4.1.part.3.name=sro_id_3@
qu.4.1.part.3.editing=useHTML@
qu.4.1.part.3.display.permute=true@
qu.4.1.part.3.answer.4=vertical asymptote of x=$a.@
qu.4.1.part.3.answer.3=horizontal asymptote of x=$a.@
qu.4.1.part.3.question=(Unset)@
qu.4.1.part.3.answer.2=vertical asymptote of y=$a.@
qu.4.1.part.3.answer.1=horizontal asymptote of y=$a.@
qu.4.1.part.3.mode=List@
qu.4.1.part.3.display=menu@
qu.4.1.part.3.credit.4=1.0@
qu.4.1.part.3.credit.3=0.0@
qu.4.1.part.3.credit.2=0.0@
qu.4.1.part.3.credit.1=0.0@
qu.4.1.question=<p>(a) Find the domain and range of $F.</p><p>(Use interval notation. Enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math> and U for union.)</p><p>Domain=<span> </span><1><span> </span></p><p>Range=<span> </span><2><span> </span></p><p><span>(b) $F has a <3><span> </span></span></p>@

qu.4.2.mode=Inline@
qu.4.2.name=Log(abs(x))DomRange@
qu.4.2.comment=<p>$plot</p>
<p>Does your answer for domain and range match the graph?</p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(0,5);
$b=rint(2,10);
$z=rint(2);
$f=switch($z,"log[($b)](abs(x-($a)))","log[($b)](abs(($a)-x))");
$fg="log[($b)](abs(x-($a)))";
$R="(-infinity,infinity)";
$D="(-infinity,$a)U($a,infinity)";
$F=maple("printf(MathML[ExportPresentation](y='$f'))");
$plot=plotmaple("plot($fg,x=-1-($a)-($b)..($a)+($b)+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.2.uid=304318be-2b38-4e3c-80df-b3cd28c8881a@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Domain and Range;
@
qu.4.2.weighting=1,1,1@
qu.4.2.numbering=alpha@
qu.4.2.part.1.name=sro_id_1@
qu.4.2.part.1.maple_answer=show("$D");@
qu.4.2.part.1.editing=useHTML@
qu.4.2.part.1.question=(Unset)@
qu.4.2.part.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.2.part.1.mode=Maple@
qu.4.2.part.1.allow2d=0@
qu.4.2.part.1.plot=@
qu.4.2.part.1.maple=grade("$RESPONSE","$D");
@
qu.4.2.part.1.type=maple@
qu.4.2.part.2.name=sro_id_2@
qu.4.2.part.2.maple_answer=show("$R");@
qu.4.2.part.2.editing=useHTML@
qu.4.2.part.2.question=(Unset)@
qu.4.2.part.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.2.part.2.mode=Maple@
qu.4.2.part.2.allow2d=0@
qu.4.2.part.2.plot=@
qu.4.2.part.2.maple=grade("$RESPONSE","$R");@
qu.4.2.part.2.type=maple@
qu.4.2.part.3.grader=exact@
qu.4.2.part.3.name=sro_id_3@
qu.4.2.part.3.editing=useHTML@
qu.4.2.part.3.display.permute=true@
qu.4.2.part.3.answer.4=vertical asymptote of x=$a.@
qu.4.2.part.3.answer.3=horizontal asymptote of x=$a.@
qu.4.2.part.3.question=(Unset)@
qu.4.2.part.3.answer.2=vertical asymptote of y=$a.@
qu.4.2.part.3.answer.1=horizontal asymptote of y=$a.@
qu.4.2.part.3.mode=List@
qu.4.2.part.3.display=menu@
qu.4.2.part.3.credit.4=1.0@
qu.4.2.part.3.credit.3=0.0@
qu.4.2.part.3.credit.2=0.0@
qu.4.2.part.3.credit.1=0.0@
qu.4.2.question=<p>(a) Find the domain and range of $F.</p><p>(Use interval notation. Enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math> and U for union.)</p><p>Domain=<span> </span><1><span> </span></p><p>Range=<span> </span><2><span> </span></p><p><span>(b) $F has a <3><span> </span></span></p>@

qu.4.3.mode=Inline@
qu.4.3.name=Exp(x)DomRange@
qu.4.3.comment=<p>$plot</p>
<p>Does your answer for domain and range match the graph?</p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$a=rint(0,5);
$f=switch(rint(2),"exp(x)+($a)","exp(-x)+($a)");
$D="(-infinity,infinity)";
$R="($a,infinity)";
$F=maple("printf(MathML[ExportPresentation](y=$f))");
$plot=plotmaple("plot($f,x=-3..3,y=-1..exp(3)+($a),thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.3.uid=ddf3c7f8-d463-4d38-8656-2ee49fdb6fab@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Domain and Range;
@
qu.4.3.weighting=1,1,1@
qu.4.3.numbering=alpha@
qu.4.3.part.1.name=sro_id_1@
qu.4.3.part.1.maple_answer=show("$D");@
qu.4.3.part.1.editing=useHTML@
qu.4.3.part.1.question=(Unset)@
qu.4.3.part.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.3.part.1.mode=Maple@
qu.4.3.part.1.allow2d=0@
qu.4.3.part.1.plot=@
qu.4.3.part.1.maple=grade("$RESPONSE","$D");
@
qu.4.3.part.1.type=maple@
qu.4.3.part.2.name=sro_id_2@
qu.4.3.part.2.maple_answer=show("$R");@
qu.4.3.part.2.editing=useHTML@
qu.4.3.part.2.question=(Unset)@
qu.4.3.part.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.3.part.2.mode=Maple@
qu.4.3.part.2.allow2d=0@
qu.4.3.part.2.plot=@
qu.4.3.part.2.maple=grade("$RESPONSE","$R");@
qu.4.3.part.2.type=maple@
qu.4.3.part.3.grader=exact@
qu.4.3.part.3.name=sro_id_3@
qu.4.3.part.3.editing=useHTML@
qu.4.3.part.3.display.permute=true@
qu.4.3.part.3.answer.4=vertical asymptote of x=$a.@
qu.4.3.part.3.answer.3=horizontal asymptote of x=$a.@
qu.4.3.part.3.question=(Unset)@
qu.4.3.part.3.answer.2=vertical asymptote of y=$a.@
qu.4.3.part.3.answer.1=horizontal asymptote of y=$a.@
qu.4.3.part.3.mode=List@
qu.4.3.part.3.display=menu@
qu.4.3.part.3.credit.4=0.0@
qu.4.3.part.3.credit.3=0.0@
qu.4.3.part.3.credit.2=0.0@
qu.4.3.part.3.credit.1=1.0@
qu.4.3.question=<p>&nbsp;</p><p>(a) Find the domain and range of $F.</p><p>(Use interval notation. Enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math> and U for union.)</p><p>Domain=<span> </span><1><span> </span></p><p>Range=<span> </span><2><span> </span></p><p><span>(b) $F has a <3><span> </span></span></p>@

qu.5.topic=Change of base@

qu.5.1.mode=Inline@
qu.5.1.name=ChangeBase@
qu.5.1.comment=@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(2,10);
$b=rint(11,20);
condition:ne(16,$b);
$q="log[$a](x)";
$q1="log[$a]($b)";
$M=maple("
MathML[ExportPresentation]('$q'),
MathML[ExportPresentation]('$q1')
");
$Q=switch(0,$M);
$Q1=switch(1,$M);
$ANS="log[$b](x)/log[$b]($a)";
$ANS1="1/log[$b]($a)";@
qu.5.1.uid=cd7bb958-9129-44ee-98c3-8164617104d9@
qu.5.1.info=  Author=Gord Clement, Jack Weiner;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Logs and Exponents;
  Sub-Topic=Change of base, logarithm;
@
qu.5.1.weighting=1,1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.name=sro_id_1@
qu.5.1.part.1.maple_answer=$ANS@
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.question=(Unset)@
qu.5.1.part.1.libname=@
qu.5.1.part.1.mode=Maple@
qu.5.1.part.1.allow2d=0@
qu.5.1.part.1.plot=@
qu.5.1.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.5.1.part.1.type=maple@
qu.5.1.part.2.name=sro_id_2@
qu.5.1.part.2.maple_answer=$ANS1@
qu.5.1.part.2.editing=useHTML@
qu.5.1.part.2.question=(Unset)@
qu.5.1.part.2.libname=@
qu.5.1.part.2.mode=Maple@
qu.5.1.part.2.allow2d=0@
qu.5.1.part.2.plot=@
qu.5.1.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.5.1.part.2.type=maple@
qu.5.1.question=<p>(a) Change $Q from base <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> to base <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math>.</p><p>&nbsp;</p><p>(Enter log[b](x) for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi mathvariant='normal'>log</mi><mrow><mi>b</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.)</p><p>&nbsp;</p><p><1></p><p>&nbsp;</p><p><span> </span></p><p><span>(b) Change $Q1 from base </span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> to base </span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math>.</span></p><p>&nbsp;</p><p><2><span> </span></p>@

qu.6.topic=LogDiff@

qu.6.1.question=<p>Given $Q, find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>.</p>@
qu.6.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.6.1.allow2d=1@
qu.6.1.maple_answer=$ANS@
qu.6.1.type=formula@
qu.6.1.mode=Maple@
qu.6.1.name=LogDiff(y=ln(fg/h))@
qu.6.1.comment=<p>$Q</p>
<p>$Step1</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>y</mi><mi>&prime;</mi></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Step2</p>@
qu.6.1.editing=useHTML@
qu.6.1.hint.1=Use log differentiation.@
qu.6.1.solution=@
qu.6.1.algorithm=$a=range(1,10);
$n=rint(2,10);
$m=switch(rint(4),"1/2","2/3","5/4","4/5");
$f=switch(rint(6),"x^($n)","sin(($a)*x)","cos(x)","tan(x)",
"ln(($a)*x)","exp(($a)*x)");
$g=switch(rint(6),"x^($n)+($a)","sin(x)","cos((($a)+1)*x)","tan(x)",
"ln(($a)*x)","exp(($a)*x)");
$h=switch(rint(6),"x^($n)-($a)","sin(x)","cos(x)","tan((($a)+2)*x)",
"ln(($a)*x)","exp(($a)*x)");
condition:ne($f,$g);
condition:ne($f,$h);
condition:ne($h,$g);
$F="ln(($f)*($g)/($h))";
$F1="ln($f)+ln($g)-ln($h)";
$M=maple("
MathML[ExportPresentation](y=$F),
MathML[ExportPresentation](y=$F1),
MathML[ExportPresentation](diff(ln($f),x)+diff(ln($g),x)-diff(ln($h),x))
");
$Q=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);
$ANS="diff(ln($f),x)+diff(ln($g),x)-diff(ln($h),x)";@
qu.6.1.uid=755375ac-d9ac-4b26-bd03-a74d1fbe5dcb@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Log differentiation;
@

qu.6.2.question=<p>Given $Q, find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>.</p>
<p>&nbsp;</p>
<p>Do not simplify (unless you really want to!)</p>@
qu.6.2.maple=evalb(simplify(($ANS1)-($RESPONSE))=0 or simplify(($ANS2)-($RESPONSE))=0);@
qu.6.2.allow2d=1@
qu.6.2.maple_answer=$ANS1 or $ANS2@
qu.6.2.type=formula@
qu.6.2.mode=Maple@
qu.6.2.name=LogDiff(y=fg/h)@
qu.6.2.comment=<p>$Q</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>y</mi></mrow></mfenced></mrow></math>&nbsp;= $Step1</p>
<p>&nbsp;</p>
<p>Differentiating both sides we obtain:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn><mrow><mi>y</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>y</mi><mi>&prime;</mi></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Step2</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>y</mi><mi>&prime;</mi></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Step3</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = $Step4.</p>@
qu.6.2.editing=useHTML@
qu.6.2.hint.1=Use Log Differentiation. If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>G</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>H</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac></mrow></math>, write your answer as either    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow><mrow><mi>F</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>G</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow><mrow><mi>G</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>H</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow><mrow><mi>H</mi></mrow></mfrac></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math> or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>G</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>H</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow><mrow><mi>F</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>G</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow><mrow><mi>G</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>H</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow><mrow><mi>H</mi></mrow></mfrac></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math>.</p>@
qu.6.2.solution=@
qu.6.2.algorithm=$a=range(1,10);
$a1=$a+1;
$a2=$a+2;
$n=rint(2,10);
$n1=$n-1;
$z1=rint(6);
$z2=rint(6);
$z3=rint(6);
$f=switch($z1,"x^($n)","sin(($a)*x)","cos(x)","tan(x)",
"ln(($a)*x)","exp(($a)*x)");
$fd=switch($z1,"x^$n","sin($a*x)","cos(x)","tan(x)",
"ln($a*x)","e^($a*x)");
$fp=switch($z1,"$n*x^$n1","$a*cos($a*x)","-sin(x)","(sec(x))^2",
"$a/x","$a*e^($a*x)");
$g=switch($z2,"x^($n)+($a)","sin(x)","cos((($a)+1)*x)","tan(x)",
"ln(($a)*x)","exp(($a)*x)");
$gd=switch($z2,"x^$n+$a","sin(x)","cos($a1*x)","tan(x)",
"ln($a*x)","e^($a*x)");
$gp=switch($z2,"$n*x^$n1","cos(x)","-($a1)*sin($a1*x)","(sec(x))^2",
"$a/x","$a*e^($a*x)");
$h=switch($z3,"x^($n)-($a)","sin(x)","cos(x)","tan((($a)+2)*x)",
"ln(($a)*x)","exp(($a)*x)");
$hd=switch($z3,"x^$n-$a","sin(x)","cos(x)","tan($a2*x)",
"ln($a*x)","e^($a*x)");
$hp=switch($z3,"$n*x^$n1","cos(x)","-sin(x)","$a2*(sec($a2*x))^2",
"$a/x","$a*e^($a*x)");
condition:ne($f,$g);
condition:ne($f,$h);
condition:ne($h,$g);
$F="($f)*($g)/($h)";
$FD="($fd)*($gd)/($hd)";

$M=maple("
MathML[ExportPresentation](y=$F),
MathML[ExportPresentation](ln($f)+ln($g)-ln($h)),
convert(y*(diff(ln($f),x)+diff(ln($g),x)-diff(ln($h),x)),string),
convert($F*(diff(ln($f),x)+diff(ln($g),x)-diff(ln($h),x)),string)
");
$Q=switch(0,$M);
$Step1=switch(1,$M);
$ANS1=switch(2,$M);
$ANS2=switch(3,$M);
$Step2=mathml("($fp)/($fd)+($gp)/($gd)-($hp)/($hd)");
$Step3=mathml("y*(($fp)/($fd)+($gp)/($gd)-($hp)/($hd))");
$Step4=mathml("($FD)*(($fp)/($fd)+($gp)/($gd)-($hp)/($hd))");@
qu.6.2.uid=acbda809-9483-42f6-9bec-33ae152562b9@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Log differentiation;
@

qu.7.topic=y=f^g@

qu.7.1.question=<p>Given $Q, find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>.</p>
<p>&nbsp;</p>
<p>Do not simplify (unless you really want to!)</p>@
qu.7.1.maple=evalb(simplify(($ANS1)-($RESPONSE))=0 or simplify(($ANS2)-($RESPONSE))=0);@
qu.7.1.allow2d=1@
qu.7.1.maple_answer=$ANS1 or $ANS2@
qu.7.1.type=formula@
qu.7.1.mode=Maple@
qu.7.1.name=LogDiff(y=f^g)@
qu.7.1.comment=<p>$Q</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Step1</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$Step2</p>
<p>Differentiating both sides we obtain,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn><mrow><mi>y</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>y</mi><mi>&prime;</mi></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Step3</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>y</mi><mi>&prime;</mi></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Step4</p>
<p>&nbsp;&nbsp;&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> $Step5</p>@
qu.7.1.editing=useHTML@
qu.7.1.hint.1=Use log differentiation. If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>F</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mrow><mi>G</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup></mrow></math>, write your answer as either    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>G</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>G</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math> or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>F</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mrow><mi>G</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>G</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>G</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math>.</p>@
qu.7.1.solution=@
qu.7.1.algorithm=$a=range(1,10);
$b=rint(1,10);
condition:ne($a,$b);
$n=rint(2,10);
$n1=$n-1;
$z1=rint(5);
$z2=rint(6);
$f=switch($z1,"x","sin(($a)*x)","cos(x)","tan(x)",
"ln(($a)*x)");
$fd=switch($z1,"x","sin($a*x)","cos(x)","tan(x)",
"ln($a*x)");
$fp=switch($z1,"1","$a*cos($a*x)","-sin(x)","(sec(x))^2",
"$a/x");
$g=switch($z2,"x^($n)+($a)","sin(x)","cos(($b)*x)","tan(x)",
"ln(($b)*x)","exp(($b)*x)");
$gd=switch($z2,"x^$n+$a","sin(x)","cos($b*x)","tan(x)",
"ln($b*x)","e^($b*x)");
$gp=switch($z2,"x^$n1","cos(x)","-$b*sin($b*x)","(sec(x))^2",
"$b/x","$b*e^($b*x)");
condition:ne($f,$g);
$F="($f)^($g)";
$FD="($fd)^($gd)";
$Q=mathml("y=$FD");
$Step1=mathml("ln($FD)");
$Step2=mathml("($gd)*ln($fd)");
$Step3=mathml("($gd)*(($fp)/($fd))+ln($fd)*($gp)");
$Step4=mathml("y*(($gd)*(($fp)/($fd))+ln($fd)*($gp))");
$Step5=mathml("($FD)*(($gd)*(($fp)/($fd))+ln($fd)*($gp))");
$M=maple("
convert(y*(($g)*diff(ln($f),x)+ln($f)*diff($g,x)),string),
convert($F*(($g)*diff(ln($f),x)+ln($f)*diff($g,x)),string)
");
$ANS1=switch(0,$M);
$ANS2=switch(1,$M);@
qu.7.1.uid=6c510833-d6e2-45c9-aad3-b8edfaceaace@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Log differentiation;
@

qu.8.topic=BasicIntFormula@

qu.8.1.question=<p>Find $F.</p>
<p>&nbsp;</p>
<p>Some of these answers will involve "ln". Remember to use absolute value when it is appropriate.</p>
<p>( <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mn>1</mn><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></math>&nbsp; while <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow><mi></mi></mrow></math> &nbsp;&nbsp; )</p>
<p>&nbsp;</p>@
qu.8.1.maple=if ((simplify(($f)-diff(($RESPONSE),x) assuming positive)=0) and (not (type(simplify(($ANS)-($RESPONSE)), numeric)))) then 1 elif ((simplify(($f)-diff(($RESPONSE),x) assuming positive)=0)) then 0.75 else 0 end if;@
qu.8.1.allow2d=1@
qu.8.1.maple_answer=ans:=$ANS:
ans+C@
qu.8.1.type=formula@
qu.8.1.mode=Maple@
qu.8.1.name=make separate fractions@
qu.8.1.comment=<p>$F</p>
<p>= $Step1</p>
<p>= $Step2</p>@
qu.8.1.editing=useHTML@
qu.8.1.hint.1=Make seperate fractions@
qu.8.1.solution=@
qu.8.1.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne(($a)*($b),0);
$n=rint(1,5);
$m=rint(6,10);
$z=rint(2);
$f=switch($z,"(x^($n)+($a)*x^($m)+($b))/x",
"(exp(($n)*x)+($b))/exp(x)");
$f1=switch($z, "x^($n-1)+($a)*x^($m-1)+($b)/x", "exp(($n-1)*x)+($b)*exp(-x)");
$ANS=switch($z,"x^($n)/($n)+($a)*x^($m)/($m)+($b)*ln(abs(x))",
"int($f,x) assuming real");
$M=maple("
MathML[ExportPresentation](Int($f,x)),
MathML[ExportPresentation](Int($f1,x)),
MathML[ExportPresentation](simplify($ANS)+C)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.8.1.uid=f58a9620-e3fd-4e5a-83e4-00a2f26020ad@
qu.8.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.8.2.question=<p>Find $F.</p>
<p>&nbsp;</p>@
qu.8.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.8.2.allow2d=1@
qu.8.2.maple_answer=$ANS+C@
qu.8.2.type=formula@
qu.8.2.mode=Maple@
qu.8.2.name=expand@
qu.8.2.comment=<p>$F</p>
<p>= $Step1</p>
<p>= $Step2</p>@
qu.8.2.editing=useHTML@
qu.8.2.hint.1=Expand@
qu.8.2.solution=@
qu.8.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne(($a)*($b),0);
condition:ne(($a)+($b),0);
$z=rint(2);
$f=switch($z,"(exp(($a)*x)+($b))^2",
"(exp(($a)*x)+($b))*(exp(($b)*x)+($a))");
$ANS=switch($z,"1/(2*($a))*exp(2*($a)*x)+2*($b)/($a)*exp(($a)*x)
+($b)^2*x","1/(($a)+($b))*exp((($a)+($b))*x)+exp(($b)*x)+
exp(($a)*x)+($a)*($b)*x");
$M=maple("
MathML[ExportPresentation](Int($f,x)),
MathML[ExportPresentation](Int(simplify(expand($f)),x)),
MathML[ExportPresentation]($ANS+C)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.8.2.uid=837e40c1-5a99-4727-a670-e76980506407@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.8.3.question=<p>Find $F.</p>@
qu.8.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.8.3.allow2d=1@
qu.8.3.maple_answer=int($f,x)+C@
qu.8.3.type=formula@
qu.8.3.mode=Maple@
qu.8.3.name=b^(ax)@
qu.8.3.comment=@
qu.8.3.editing=useHTML@
qu.8.3.solution=@
qu.8.3.algorithm=$a=rint(-5,5);
$b=rint(2,10);
condition:ne($a,0);
$f=switch(rint(2),"exp(($a)*x)","($b)^(($a)*x)");
$F=maple("printf(MathML[ExportPresentation](Int($f,x)))");@
qu.8.3.uid=01bf7705-da22-4751-a518-cb714d9d3f78@
qu.8.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.9.topic=CRIR, f(exp),f(ln)@

qu.9.1.question=<p>Find $F.</p>
<p>&nbsp;</p>@
qu.9.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.9.1.allow2d=1@
qu.9.1.maple_answer=$ANS+C@
qu.9.1.type=formula@
qu.9.1.mode=Maple@
qu.9.1.name=CRIR NoAdjustments,f(exp(x))@
qu.9.1.comment=<p>Note $Step1 is the derivative of $Step2. Now do you see the chain rule in reverse?</p>@
qu.9.1.editing=useHTML@
qu.9.1.hint.1=This question is an example of the Chain Rule in reverse, with no "adjustments" needed.    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>g</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>f</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math></p>@
qu.9.1.solution=@
qu.9.1.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=rint(2,10);
$z=rint(5);
$f=switch($z,"x^($m)","sin(x)","cos(x)",
"sec(x)^2","csc(x)^2");
$g=switch(rint(2),"exp(($a)*x)+($m)",
"exp(x)+($a)");
$fg="subs(x=($g),$f)";
$gp="diff($g,x)";
$M=maple("
MathML[ExportPresentation](Int($gp*$fg,x)),
MathML[ExportPresentation]($gp),
MathML[ExportPresentation]($g)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);
$ANS=switch($z,"($g)^(($m)+1)/(($m)+1)","-cos($g)","sin($g)",
"tan($g)","-cot($g)");@
qu.9.1.uid=3613d395-e855-4fdd-9113-4b42f4235fa7@
qu.9.1.info=  Author=Jack Weiner, Gord Clemet;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.9.2.question=<p>Find $F.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.9.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.9.2.allow2d=1@
qu.9.2.maple_answer=$ANS+C@
qu.9.2.type=formula@
qu.9.2.mode=Maple@
qu.9.2.name=CRIR Adjustments,f(ln(x))@
qu.9.2.comment=<p>Note the $Step2 is the derivative of $Step1. Now do you see the chain rule in reverse?</p>@
qu.9.2.editing=useHTML@
qu.9.2.hint.1=This question is an example of the Chain Rule in reverse, but you will need to adjust the integrand with a multiplicative constant. For example,    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfenced open='(' close=')' separators=','><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mn>4</mn><mrow><mn>21</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math>.</p>@
qu.9.2.solution=@
qu.9.2.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=rint(2,10);
$n=rint(2,10);
$z=rint(5);
$f=switch($z,"x^($m)","sin(x)","cos(x)","sec(x)^2","csc(x)^2");
$g=switch(rint(2),"ln(($n)*x+$m)",
"ln(($n)*x-($m))");
$fg="subs(x=($g),$f)";
$gp="diff($g,x)";
$Q="($gp)*($fg)/($n)";
$ANS=switch($z,"1/($n)*($g)^(($m)+1)/(($m)+1)","-1/($n)*cos($g)",
"1/($n)*sin($g)","(1/($n))*tan(($g))","-(1/($n))*cot($g)");
$M=maple("
MathML[ExportPresentation](Int(($gp)*($fg)/($n),x)),
MathML[ExportPresentation]($g),
MathML[ExportPresentation]($gp)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.9.2.uid=f792f980-e322-4cf2-9167-a00aa17d5ccb@
qu.9.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.9.3.question=<p>Find $F.</p>
<p>&nbsp;</p>@
qu.9.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.9.3.allow2d=1@
qu.9.3.maple_answer=$ANS+C@
qu.9.3.type=formula@
qu.9.3.mode=Maple@
qu.9.3.name=CRIR NoAdjustments,f(ln(x))@
qu.9.3.comment=<p>Note that $Step1 is the derivative of $Step2. Now do you see the chain rule in reverse?</p>@
qu.9.3.editing=useHTML@
qu.9.3.hint.1=This question is an example of the Chain Rule in reverse, with no "adjustments" needed.    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>g</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>f</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math></p>@
qu.9.3.solution=@
qu.9.3.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=rint(2,10);
$z=rint(5);
$f=switch($z,"x^($m)","sin(x)","cos(x)",
"sec(x)^2","csc(x)^2");
$g=switch(rint(2),"ln(x)","ln(($a)*x+($m))");
$fg="subs(x=($g),$f)";
$gp="diff($g,x)";
$M=maple("
MathML[ExportPresentation](Int($gp*$fg,x)),
MathML[ExportPresentation]($gp),
MathML[ExportPresentation]($g)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);
$ANS=switch($z,"($g)^(($m)+1)/(($m)+1)","-cos($g)","sin($g)",
"tan($g)","-cot($g)");@
qu.9.3.uid=a412e616-f858-427c-8085-3ccca4a1d4a9@
qu.9.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.9.4.question=<p>Find $F.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.9.4.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.9.4.allow2d=1@
qu.9.4.maple_answer=$ANS+C@
qu.9.4.type=formula@
qu.9.4.mode=Maple@
qu.9.4.name=CRIR Adjustments,f(exp(x))@
qu.9.4.comment=<p>Note that $Step1 is the derivative of $Step2. Now do you see the chain rule in reverse?</p>@
qu.9.4.editing=useHTML@
qu.9.4.hint.1=This question is an example of the Chain Rule in reverse, but you will need to adjust the integrand with a multiplicative constant. For example,    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfenced open='(' close=')' separators=','><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mn>4</mn><mrow><mn>21</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math>.</p>@
qu.9.4.solution=@
qu.9.4.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=rint(2,10);
$n=rint(2,10);
$z=rint(4);
$f=switch($z,"sin(x)","cos(x)","sec(x)^2","csc(x)^2");
$g=switch(rint(2),"exp(($n)*x)+($m)","exp(($n)*x-($m))");
$fg="subs(x=($g),$f)";
$gp="diff($g,x)";
$Q="($gp)*($fg)/($n)";
$ANS=switch($z,"-1/($n)*cos($g)",
"1/($n)*sin($g)","(1/($n))*tan(($g))","-(1/($n))*cot($g)");
$M=maple("
MathML[ExportPresentation](Int(($gp)*($fg)/($n),x)),
MathML[ExportPresentation]($gp),
MathML[ExportPresentation]($g)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.9.4.uid=e798bf26-28e5-467c-8e9e-a3e91ef4eaa8@
qu.9.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.10.topic=CRIR,exp(f),u'/u@

qu.10.1.question=<p>Find $F.</p>
<p>&nbsp;</p>
<p>This question involves "ln". Use "abs" for absolute value only if it is really necessary.</p>
<p>&nbsp;</p>@
qu.10.1.maple=if((not(depends(simplify($ANS-$RESPONSE),x))) and (not(type(simplify($ANS-$RESPONSE),numeric)))) then 1 elif((not(depends(simplify($ANS-$RESPONSE),x)))) then 0.75  else 0 end if;@
qu.10.1.allow2d=1@
qu.10.1.maple_answer=$ANS+C@
qu.10.1.type=formula@
qu.10.1.mode=Maple@
qu.10.1.name=CRIR NoAdjustments,ln(f(x)@
qu.10.1.comment=<p>Note $Step1 is the derivative of $Step2. Now do you see the chain rule in reverse?</p>@
qu.10.1.editing=useHTML@
qu.10.1.hint.1=This question is an example of the Chain Rule in reverse, with no "adjustments" needed.    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>g</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>f</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math>.</p>@
qu.10.1.solution=@
qu.10.1.algorithm=$a=rint(2,5);
$m=rint(3,9,2);
$p=rint(2,10,2);
$n=rint(2,10);
$z=rint(7);
$g=switch($z,"($a)*x^($p)+($n)","($a)*x^($m)+($n)","sin(($a)*x)","cos(($a)*x)",
"sec(($a)*x)","csc(($a)*x)","exp(($a)*x)+($m)");
$fg="subs(x=($g),1/x)";
$gp="diff($g,x)";
$ANS=switch($z,"ln(($a)*x^($p)+($n))","ln(abs(($a)*x^($m)+($n)))",
"ln(abs(sin(($a)*x)))","ln(abs(cos(($a)*x)))",
"ln(abs(sec(($a)*x)))","ln(abs(csc(($a)*x)))","ln(exp(($a)*x)+($m))");
$M=maple("
MathML[ExportPresentation](Int(($gp)*($fg),x)),
MathML[ExportPresentation]($gp),
MathML[ExportPresentation]($g)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.10.1.uid=b751168a-e4c7-40bf-9f58-995612745bdd@
qu.10.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.10.2.question=<p>Find $F.</p>
<p>&nbsp;</p>@
qu.10.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.10.2.allow2d=1@
qu.10.2.maple_answer=$ANS+C@
qu.10.2.type=formula@
qu.10.2.mode=Maple@
qu.10.2.name=CRIR Adjustments,exp(f(x)@
qu.10.2.comment=<p>Note $Step1 is the derivative of $Step2. Now do you see the chain rule in reverse?</p>@
qu.10.2.editing=useHTML@
qu.10.2.hint.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>g</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>f</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math>@
qu.10.2.solution=@
qu.10.2.algorithm=$a=rint(1,5);
$m=rint(2,10);
$n=rint(2,10);
$z=rint(6);
$g=switch($z,"($a)*x^($m)+($n)","sin(($a)*x)","cos(($a)*x)",
"sec(($a)*x)","csc(($a)*x)","exp(($a)*x)");
$fg="subs(x=($g),exp(x))";
$gp="diff($g,x)";
$Q="($gp)*($fg)/($a)";
$ANS="int(($gp)*($fg)/($a),x)";
$M=maple("
MathML[ExportPresentation](Int(($gp)*($fg)/($a),x)),
MathML[ExportPresentation]($gp),
MathML[ExportPresentation]($g)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.10.2.uid=d2f9b6c2-8ef7-4145-873d-78ce6e9f1cf9@
qu.10.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.10.3.question=<p>Find $F.</p>
<p>&nbsp;</p>
<p>This question involves "ln". Use "abs" for absolute value only if it is really necessary.</p>
<p>&nbsp;</p>@
qu.10.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.10.3.allow2d=1@
qu.10.3.maple_answer=$ANS+C@
qu.10.3.type=formula@
qu.10.3.mode=Maple@
qu.10.3.name=CRIR NoAdjustments,exp(f(x)@
qu.10.3.comment=<p>Note $Step1 is the derivative of $Step2. Now do you see the chain rule in reverse?</p>@
qu.10.3.editing=useHTML@
qu.10.3.hint.1=<p>This question is an example of the Chain Rule in reverse, with no "adjustments" needed.</p>    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>g</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>g</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></math>.</p>@
qu.10.3.solution=@
qu.10.3.algorithm=$a=rint(1,5);
$m=rint(2,10);
$n=rint(2,10);
$z=rint(6);
$g=switch($z,"($a)*x^($m)+($n)","sin(($a)*x)","cos(($a)*x)",
"sec(($a)*x)","csc(($a)*x)","exp(($a)*x)");
$fg="subs(x=($g),exp(x))";
$gp="diff($g,x)";
$ANS=switch($z,"exp(($a)*x^($m)+($n))","exp(sin(($a)*x))","exp(cos(($a)*x))", "exp(sec(($a)*x))","exp(csc(($a)*x))","exp(exp(($a)*x))");
$M=maple("
MathML[ExportPresentation](Int(($gp)*($fg),x)),
MathML[ExportPresentation]($gp),
MathML[ExportPresentation]($g)
");

$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.10.3.uid=0afae7d6-6433-4c66-9a3f-0b761889465d@
qu.10.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Integration;
@

qu.10.4.question=<p>Find $F.</p>
<p>&nbsp;</p>
<p>This question involves "ln". Use "abs" for absolute value only if it is really necessary.</p>
<p>&nbsp;</p>@
qu.10.4.maple=if((not(depends(simplify($ANS-$RESPONSE),x))) and (not(type(simplify($ANS-$RESPONSE),numeric)))) then 1 elif((not(depends(simplify($ANS-$RESPONSE),x)))) then 0.75  else 0 end if;@
qu.10.4.allow2d=1@
qu.10.4.maple_answer=$ANS+C@
qu.10.4.type=formula@
qu.10.4.mode=Maple@
qu.10.4.name=CRIR Adjustments,ln(f(x)@
qu.10.4.comment=<p>Note $Step1 is the derivative of $Step2. Now do you see the chain rule in reverse?</p>@
qu.10.4.editing=useHTML@
qu.10.4.hint.1=This question is an example of the Chain Rule in reverse, with "adjustments" needed. Or not.    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>g</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>f</mi><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.0em'>&amp;apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math>.</p>@
qu.10.4.solution=@
qu.10.4.algorithm=$a=rint(2,7);
$m=rint(3,9,2);
$p=rint(2,10,2);
$n=rint(2,10);
$z=rint(7);
$g=switch($z,"($a)*x^($p)+($n)","($a)*x^($m)+($n)","sin(($a)*x)","cos(($a)*x)",
"sec(($a)*x)","csc(($a)*x)","exp(($a)*x)+($m)");
$fg="subs(x=($g),1/x)";
$gp="diff($g,x)";
$Q="($gp)*($fg)/($a)";
$ANS=switch($z,"1/($a)*ln(($a)*x^($p)+($n))","1/($a)*ln(abs(($a)*x^($m)+($n)))",
"1/($a)*ln(abs(sin(($a)*x)))","1/($a)*ln(abs(cos(($a)*x)))",
"1/($a)*ln(abs(sec(($a)*x)))","1/($a)*ln(abs(csc(($a)*x)))",
"1/($a)*ln(exp(($a)*x)+($m))");

$M=maple("
MathML[ExportPresentation](Int(($gp)*($fg)/($a),x)),
MathML[ExportPresentation]($gp),
MathML[ExportPresentation]($g)
");
$F=switch(0,$M);
$Step1=switch(1,$M);
$Step2=switch(2,$M);@
qu.10.4.uid=67872851-e3ab-4ba5-abc9-102962c20857@
qu.10.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponenets;
  Sub-Topic=Integration;
@

qu.11.topic=Theory@

qu.11.1.mode=Multiple Selection@
qu.11.1.name=theory@
qu.11.1.comment=@
qu.11.1.editing=useHTML@
qu.11.1.solution=@
qu.11.1.algorithm=@
qu.11.1.uid=e1e36000-45d8-404d-b0aa-1852ef0ea379@
qu.11.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Logs and Exponents;
  Sub-Topic=Theory;
@
qu.11.1.question=<p>Click beside each TRUE statement.</p>@
qu.11.1.answer=1, 2, 3, 4, 5@
qu.11.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>b</mi><mrow><mi>x</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mn>0</mn></mrow></munder><mfrac><mfenced open='(' close=')' separators=','><mrow><msup><mi>b</mi><mrow><mi>h</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi>h</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>b</mi><mrow><mi>x</mi></mrow></msup></mrow></math>@
qu.11.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>b</mi><mrow><mi>x</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>b</mi><mrow><mi>x</mi></mrow></msup></mrow></math>@
qu.11.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mfrac><mn>1</mn><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced><mrow><mi>n</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>e</mi></mrow></math>@
qu.11.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mn>0</mn></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>h</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mrow><mi>h</mi></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>e</mi></mrow></math>@
qu.11.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo><mfenced open='|' close='|' separators=','><mrow><mi>x</mi></mrow></mfenced><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mrow><mfrac><mn>1</mn><mrow><mi>x</mi></mrow></mfrac></mrow></mrow></math>@
qu.11.1.fixed=@

